The number is called the order of the Bessel equation; in the general case and assume complex values. The substitution yields the reduced form of equation (1):

(2)

A Bessel equation is a special case of a confluent hypergeometric equation; if is substituted into (2), equation (2) becomes a Whittaker equation. In equation (1) the point is weakly singular, while the point is strongly singular. For this reason a Bessel equation does not belong to the class of Fuchsian equations (cf. Fuchsian equation). F. Bessel [1] was the first to study equation (1) systematically, but such equations are encountered even earlier in the works of D. Bernoulli, L. Euler and J.L. Lagrange.

A Bessel equation results from separation of variables in many problems of mathematical physics [2], particularly in the case of boundary value problems of potential theory for a cylindrical domain.

The solutions of Bessel equations are called cylinder functions (or Bessel functions). These may be subdivided into the cylinder functions of the first kind (Bessel functions) , the cylinder functions of the second kind (Weber functions or Neumann functions, (cf. Weber function; Neumann function) and the cylinder functions of the third kind (Hankel functions) , . If the order is fixed, all these functions are analytic functions of the complex argument ; for all these functions, except for the functions of integer order, the point is a branch point. If the argument is fixed, all these functions are single-valued entire functions of the complex order [3].

If the order is not an integer, then the general solution of equation (1) may be written as

where are arbitrary constants. For a given order, any two of the functions , , , are linearly independent and may serve as a fundamental system of solutions of (1). For this reason, the general solution of equation (1) can be represented, in particular, in the following forms:

The following equations are closely connected with equation (1): the equation

which becomes (1) as a result of the substitution , and with as a fundamental system of solutions the modified cylinder functions (Bessel functions of imaginary argument), and the equation

which becomes equation (1) as a result of the substitution and which has the Kelvin functions as its fundamental system of solutions. Many other second-order linear ordinary differential equations (e.g. the Airy equation) can also be transformed into equation (1) by a transformation of the unknown function and the independent variable. The solution of a series of linear equations of higher orders may be written in the form of Bessel functions [4].